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where μ= cos e, and i) represents any positive integer whatever, provided ¿ is never greater than ¿r+1) ̧

Though we have thus the solution of every equation in the system (41), yet that of the first may be obtained under a simpler form by writing therein for X, its value —¿(2) deduced from (45). We shall then immediately perceive that it is satisfied by

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In consequence of the formula (45), the equation (42) becomes

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+

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— λ1 — (¿ (®) + 2w) (¿ (8) +2w+n−1),

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2w.2w - 2 × 2¿(s) + 2w+s−2.2i(®) + 2w+s-4
2.4 × 22+ 4w + n + 3. 2i + 4w+ n − 5

where a represents any whole positive number.

Having thus determined all the factors of p,

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remains to deduce the corresponding value of H. But à ̧ the particular value satisfying the differential equation in H, will be had from by simply making therein

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since in the present case we have generally a, a'.

=

Hence, it is clear that the proper values of 0,, 0, 0, &c. to be here employed are all constant, and consequently the factor

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entering into is likewise constant. Neglecting therefore this factor as superfluous, we get for the particular value of H,

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and P. represents what P becomes when p is changed into.

Substituting this value of H, in the equation (25), No. 10, there results since a2= a+h3

H = K. P2 |

hdh

.(46),

'P2 (a'2 + h2)3

K being an arbitrary constant quantity.

Thus the complete value of V for the particular case considered in the present number is

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and the equation (27), No. 11, will give for the corresponding

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where P',

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',,, &c. are the values which the functions P, ,,,, &c. take when we change the unaccented variables ,,,,..., into the corresponding accented ones, §......

and

P1 =

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n+2i+2w − 1. n + 2i+2w+1...... n + 2i + 4w-3'

-

or the value of P when p=1; where as well as in what follows

i is written in the place of ").

The differential equation which serves to determine H when we introduce a instead of h as independent variable, may in the present case be written under the form

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+ {i (i + 8 − 2) a” — (i + 2w) (i + 2w+ n − 1) a3} H,

and the particular integral here required is that which vanishes when his infinite. Moreover it is easy to prove, by expanding in series, that this particular integral is

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H= k'a1Ó.a” „a12r-•-a da (aa — a'2) ;

provided we make the variable r to which A" refers vanish after all the operations have been effected.

But the constant may be determined by comparing the coefficient of the highest power of a in the expansion of the last formula with the like coefficient in that of the expression (46), and thus we have

= Ka'12 ( − 1 )∞ 2 + 2i+2w −1.n+2i+2w+1 ... n + 2i+4w−3

k=

'+200

2.4.6 2w

Hence we readily get for the equivalent of (47),

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n+2i+2w-1.n+2i+2w+1...n+2i+4w-3 2.4.6...20

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s„daa”

In certain cases the value of Vjust obtained will be found more convenient than the foregoing one (47). Suppose for instance we represent the value of V when h=0, or a=a' by V。. Then we shall hence get

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n+2i+2w-1.n+2i+2w+1...n+2i+4w-3 2.4.6...20

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since in the formula (8), r ought to be made equal to zero at the end of the process.

By conceiving the auxiliary variable u to vanish, it will become clear from what has been advanced in the preceding number, that the values of the function V within circular planes and spheres are only particular cases of the more general one (49), which answer to s=2 and s=3 respectively. We have thus by combining the expressions (48) and (49), the means of determining V, when the density p' is given, and vice versa; and the present method of resolving these problems seems more simple if possible than that contained in the articles (4) and (5) former paper.

of my

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* From the Transactions of the Cambridge Philosophical Society, 1838. [Read May 15, 1837.]

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